Bài 1 trang 40 SGK Toán 11 tập 1 - Cánh diều

Đề bài

Giải phương trình:

a)     $\sin \left(2x-\frac{\pi }{3}\right)=-\frac{\sqrt{3}}{2}$

b)     $\sin \left(3x+\frac{\pi }{4}\right)=-\frac{1}{2}$

c)     $\cos \left(\frac{x}{2}+\frac{\pi }{4}\right)=\frac{\sqrt{3}}{2}$

d)     $2\cos 3x+5=3$

e)     $3\tan x=-\sqrt{3}$

g)      $\cot x-3=\sqrt{3}\left(1-\cot x\right)$

Lời giải chi tiết

a)     $\sin \left(2x-\frac{\pi }{3}\right)=-\frac{\sqrt{3}}{2}$

$\begin{array}{l}\iff [\begin{array}{c}2x-\frac{\pi }{3}=-\frac{\pi }{3}+k2\pi \\ 2x-\frac{\pi }{3}=\pi +\frac{\pi }{3}+k2\pi \end{array}\left(k\in \mathbb{Z}\right)\\ \iff [\begin{array}{c}2x=k2\pi \\ 2x=\frac{5\pi }{3}+k2\pi \end{array}\left(k\in \mathbb{Z}\right)\\ \iff [\begin{array}{c}x=k\pi \\ x=\frac{5\pi }{6}+k\pi \end{array}\left(k\in \mathbb{Z}\right)\end{array}$

Vậy phương trình có nghiệm là: $x\in \left\{k\pi ;\frac{5\pi }{6}+k\pi \right\}$

b)     $\sin \left(3x+\frac{\pi }{4}\right)=-\frac{1}{2}$

$\begin{array}{l}\iff [\begin{array}{c}3x+\frac{\pi }{4}=-\frac{\pi }{6}+k2\pi \\ 3x+\frac{\pi }{4}=\frac{7\pi }{6}+k2\pi \end{array}\left(k\in \mathbb{Z}\right)\\ \iff [\begin{array}{c}3x=-\frac{5\pi }{12}+k2\pi \\ 3x=\frac{11\pi }{12}+k2\pi \end{array}\left(k\in \mathbb{Z}\right)\\ \iff [\begin{array}{c}x=-\frac{5\pi }{36}+k\frac{2\pi }{3}\\ x=\frac{11\pi }{36}+k\frac{2\pi }{3}\end{array}\left(k\in \mathbb{Z}\right)\end{array}$

c)     $\cos \left(\frac{x}{2}+\frac{\pi }{4}\right)=\frac{\sqrt{3}}{2}$

$\begin{array}{l}\iff [\begin{array}{c}\frac{x}{2}+\frac{\pi }{4}=\frac{\pi }{6}+k2\pi \\ \frac{x}{2}+\frac{\pi }{4}=-\frac{\pi }{6}+k2\pi \end{array}\left(k\in \mathbb{Z}\right)\\ \iff [\begin{array}{c}\frac{x}{2}=-\frac{\pi }{12}+k2\pi \\ \frac{x}{2}=-\frac{5\pi }{12}+k2\pi \end{array}\left(k\in \mathbb{Z}\right)\\ \iff [\begin{array}{c}x=-\frac{\pi }{6}+k4\pi \\ x=-\frac{5\pi }{6}+k4\pi \end{array}\left(k\in \mathbb{Z}\right)\end{array}$

d)     $2\cos 3x+5=3$

$\begin{array}{l}\iff \cos 3x=-1\\ \iff [\begin{array}{c}3x=\pi +k2\pi \\ 3x=-\pi +k2\pi \end{array}\left(k\in \mathbb{Z}\right)\\ \iff [\begin{array}{c}x=\frac{\pi }{3}+k\frac{2\pi }{3}\\ x=\frac{-\pi }{3}+k\frac{2\pi }{3}\end{array}\left(k\in \mathbb{Z}\right)\end{array}$

e)      

$\begin{array}{l}3\tan x=-\sqrt{3}\\ \iff \tan x=\frac{-\sqrt{3}}{3}\\ \iff \tan x=\tan \left(-\frac{\pi }{6}\right)\\ \iff x=-\frac{\pi }{6}+k\pi \end{array}$

g)

$\begin{array}{l}\cot x-3=\sqrt{3}\left(1-\cot x\right)\\ \iff \cot x-3=\sqrt{3}-\sqrt{3}\cot x\\ \iff \cot x+\sqrt{3}\cot x=\sqrt{3}+3\\ \iff (1+\sqrt{3})\cot x=\sqrt{3}+3\\ \iff \cot x=\sqrt{3}\\ \iff \cot x=\cot \frac{\pi }{6}\\ \iff x=\frac{\pi }{6}+k\pi \end{array}$